I'd like to show that the Euler characteristic ($V – E + F$) of a compact oriented surface without boundary, $S$$g$, is of the form $2 – 2g$ where $S$$g$ is a sphere with $g$ handles.
A sphere with handles is obtained by cutting $2g$ disks out of the sphere and gluing in $g$ cylinders along the boundary circles. Hence, $S$$0$ is the sphere and the torus is $S$$1$.
I'm pretty new to topology. But I think induction would make for an easy, understandable proof where the base case is the sphere.
Any help or solutions are appreciated!
Best Answer
You could represent your surface with $g$ handles as a polygon with $4g$ edges suitably identified in pairs, and then compute $V$, $E$, and $F$ for that polygon, and then $V-E+F$.